{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-advance-20261004-r06-system-linear-equations-0085",
  "pilot_number": "R0064-32985",
  "number": 32985,
  "family_id": "qf-linear-system-unique-solve",
  "title": {
    "zh-Hant": "第 32985 題：非零行列式的精確唯一解",
    "en": "Question 32985: Exact Unique Solution with a Nonzero Determinant"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-advance-20261004-r06-system-linear-equations-0085",
    "batch_id": "batch-20261003-advance-20261004-r06-system-linear-equations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-system-linear-equations"
    ],
    "title_zh": "非零行列式的精確唯一解",
    "stem_zh": "實數論域中兩個限制 (1)x+(1)y=-3 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。",
    "answer": {
      "kind": "expression",
      "value": "(-4/3, -5/3)",
      "explanation_zh": "D=-3；解=(-4/3, -5/3)。以精確有理列消去得到兩個主元與解，不使用浮點交點；D=0 的系統不套此唯一解題型。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "非零行列式的精確唯一解"
      ]
    },
    "variant": {
      "family_id": "qf-linear-system-unique-solve",
      "seed": 202610046085,
      "parameters": {
        "rows": [
          [
            1,
            1,
            -3
          ],
          [
            1,
            -2,
            2
          ]
        ],
        "domain": "reals",
        "operation": "unique_solve",
        "semantic_template": "qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-3\"]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-advance-20261004-r06-system-linear-equations-0085",
    "batch_id": "batch-20261003-advance-20261004-r06-system-linear-equations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3404b18894eba0da8ac3789666cc260eacfd8e48b8c2aadcc9eef1bb02ba464b",
      "family_id": "qf-linear-system-unique-solve",
      "seed": 202610046085,
      "semantic_template": "qf-linear-system-unique-solve/{\"domain\":\"reals\",\"operation\":\"unique_solve\",\"rows\":[[\"1\",\"-2\",\"2\"],[\"1\",\"1\",\"-3\"]]}"
    },
    "fields": {
      "title": "Exact Unique Solution with a Nonzero Determinant",
      "stem": "Over the reals, the constraints (1)x+(1)y=-3 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.",
      "answer": {
        "kind": "expression",
        "value": "(-4/3, -5/3)",
        "explanation": "D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-4/3, -5/3). The D=0 cases are outside this unique-solution exercise."
      },
      "skills": [
        "Exact Unique Solution with a Nonzero Determinant"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "實數論域中兩個限制 (1)x+(1)y=-3 與 (1)x+(-2)y=2 必須同時成立，且 D≠0。求精確有理數解 (x,y)，各分量採最簡正分母 p/q。",
        "en": "Over the reals, the constraints (1)x+(1)y=-3 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-linear-system-unique-solve, unique_solve, reals, []]",
      "after": "derived = [[solution_pair_fractions_RETAIN1]]",
      "relation": "implies",
      "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。",
        "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[[1, 1, -3], [1, -2, 2]]]",
      "after": "derived = [[coefficient_x, coefficient_y, right_side]]",
      "relation": "implies",
      "rule_id": "System_RAW_exact2_augmented_rows3_ALL_SAFEincl_unused_rhs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原兩row各[a,b,c]代表ax+by=c，第三欄是rhs，不是矩陣係數。全部6個raw都SAFE先驗，D題未用rhs也不能略過；原row次序不改。",
        "en": "Each ORIGINAL row[a,b,c]means ax+by=c;the THIRD column is RHS,not a coefficient.Type ALL SIX raw SAFE values,including unused RHS in a determinant task.Keep original row order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "System ORIGINAL a times e main product := mul(1, -2)",
      "after": "System ORIGINAL a times e main product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 System ORIGINAL a times e main product，不以小數近似替代。",
          "en": "Recompute System ORIGINAL a times e main product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "System ORIGINAL a times e main product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "System ORIGINAL b times d other product := mul(1, 1)",
      "after": "System ORIGINAL b times d other product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 System ORIGINAL b times d other product，不以小數近似替代。",
          "en": "Recompute System ORIGINAL b times d other product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "System ORIGINAL b times d other product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "System ORIGINAL signed determinant ae minus bd := sub(-2, 1)",
      "after": "System ORIGINAL signed determinant ae minus bd = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 System ORIGINAL signed determinant ae minus bd，不以小數近似替代。",
          "en": "Recompute System ORIGINAL signed determinant ae minus bd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-3",
        "label": "System ORIGINAL signed determinant ae minus bd"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [[[1, 1, -3], [1, -2, 2]]]",
      "after": "derived = [-2, 1, -3]",
      "relation": "implies",
      "rule_id": "System_DETERMINANT_coefficients_ONLY_raw_order_sign",
      "conditions": [],
      "explanation": {
        "zh-Hant": "D=ae−bd只取兩row前兩欄，保原row順序與符號；rhs不算入，row交換會改D符號，不用semantic排序代替原資料。",
        "en": "D=ae−bd uses ONLY the first TWO columns,retaining ORIGINAL row order/sign.RHS is excluded;swapping rows changes the sign.Do not replace originals by semantic sorting."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Cramer ORIGINAL c times e := mul(-3, -2)",
      "after": "Cramer ORIGINAL c times e = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL c times e，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL c times e with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "-2"
        ],
        "output": "6",
        "label": "Cramer ORIGINAL c times e"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Cramer ORIGINAL b times f := mul(1, 2)",
      "after": "Cramer ORIGINAL b times f = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL b times f，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL b times f with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Cramer ORIGINAL b times f"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Cramer ORIGINAL numerator x ce minus bf := sub(6, 2)",
      "after": "Cramer ORIGINAL numerator x ce minus bf = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL numerator x ce minus bf，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL numerator x ce minus bf with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "2"
        ],
        "output": "4",
        "label": "Cramer ORIGINAL numerator x ce minus bf"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Cramer ORIGINAL a times f := mul(1, 2)",
      "after": "Cramer ORIGINAL a times f = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL a times f，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL a times f with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Cramer ORIGINAL a times f"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Cramer ORIGINAL c times d := mul(-3, 1)",
      "after": "Cramer ORIGINAL c times d = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL c times d，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL c times d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "Cramer ORIGINAL c times d"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Cramer ORIGINAL numerator y af minus cd := sub(2, -3)",
      "after": "Cramer ORIGINAL numerator y af minus cd = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL numerator y af minus cd，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL numerator y af minus cd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "-3"
        ],
        "output": "5",
        "label": "Cramer ORIGINAL numerator y af minus cd"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Cramer ORIGINAL x numerator over NONZERO D := div(4, -3)",
      "after": "Cramer ORIGINAL x numerator over NONZERO D = -4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL x numerator over NONZERO D，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL x numerator over NONZERO D with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "-3"
        ],
        "output": "-4/3",
        "label": "Cramer ORIGINAL x numerator over NONZERO D"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Cramer ORIGINAL y numerator over NONZERO D := div(5, -3)",
      "after": "Cramer ORIGINAL y numerator over NONZERO D = -5/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer ORIGINAL y numerator over NONZERO D，不以小數近似替代。",
          "en": "Recompute Cramer ORIGINAL y numerator over NONZERO D with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "-3"
        ],
        "output": "-5/3",
        "label": "Cramer ORIGINAL y numerator over NONZERO D"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Cramer reverse ORIGINAL row0 x term := mul(1, -4/3)",
      "after": "Cramer reverse ORIGINAL row0 x term = -4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row0 x term，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row0 x term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-4/3"
        ],
        "output": "-4/3",
        "label": "Cramer reverse ORIGINAL row0 x term"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Cramer reverse ORIGINAL row0 y term := mul(1, -5/3)",
      "after": "Cramer reverse ORIGINAL row0 y term = -5/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row0 y term，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row0 y term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-5/3"
        ],
        "output": "-5/3",
        "label": "Cramer reverse ORIGINAL row0 y term"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Cramer reverse ORIGINAL row0 terms sum := sum(-4/3, -5/3)",
      "after": "Cramer reverse ORIGINAL row0 terms sum = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row0 terms sum，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row0 terms sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4/3",
          "-5/3"
        ],
        "output": "-3",
        "label": "Cramer reverse ORIGINAL row0 terms sum"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Cramer reverse ORIGINAL row0 residual minus rhs := sub(-3, -3)",
      "after": "Cramer reverse ORIGINAL row0 residual minus rhs = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row0 residual minus rhs，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row0 residual minus rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "-3"
        ],
        "output": "0",
        "label": "Cramer reverse ORIGINAL row0 residual minus rhs"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [Cramer reverse, 0, 1, 1, -3, -4/3, -5/3]",
      "after": "derived = [-4/3, -5/3, -3, 0/1]",
      "relation": "implies",
      "rule_id": "System_EACH_original_row_reverse_exact_substitution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原row以原係數及rhs回代，全部殘差0才解整個系統，不是只驗整理過的一列。",
        "en": "Substitute into EACH ORIGINAL row with its ORIGINAL coefficients/RHS.ALL residuals must be0,not merely one transformed row."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Cramer reverse ORIGINAL row1 x term := mul(1, -4/3)",
      "after": "Cramer reverse ORIGINAL row1 x term = -4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row1 x term，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row1 x term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-4/3"
        ],
        "output": "-4/3",
        "label": "Cramer reverse ORIGINAL row1 x term"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Cramer reverse ORIGINAL row1 y term := mul(-2, -5/3)",
      "after": "Cramer reverse ORIGINAL row1 y term = 10/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row1 y term，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row1 y term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-5/3"
        ],
        "output": "10/3",
        "label": "Cramer reverse ORIGINAL row1 y term"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Cramer reverse ORIGINAL row1 terms sum := sum(-4/3, 10/3)",
      "after": "Cramer reverse ORIGINAL row1 terms sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row1 terms sum，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row1 terms sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4/3",
          "10/3"
        ],
        "output": "2",
        "label": "Cramer reverse ORIGINAL row1 terms sum"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Cramer reverse ORIGINAL row1 residual minus rhs := sub(2, 2)",
      "after": "Cramer reverse ORIGINAL row1 residual minus rhs = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cramer reverse ORIGINAL row1 residual minus rhs，不以小數近似替代。",
          "en": "Recompute Cramer reverse ORIGINAL row1 residual minus rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "Cramer reverse ORIGINAL row1 residual minus rhs"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [Cramer reverse, 1, 1, -2, 2, -4/3, -5/3]",
      "after": "derived = [-4/3, 10/3, 2, 0/1]",
      "relation": "implies",
      "rule_id": "System_EACH_original_row_reverse_exact_substitution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原row以原係數及rhs回代，全部殘差0才解整個系統，不是只驗整理過的一列。",
        "en": "Substitute into EACH ORIGINAL row with its ORIGINAL coefficients/RHS.ALL residuals must be0,not merely one transformed row."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [[[1, 1, -3], [1, -2, 2]], -3]",
      "after": "derived = [4, 5, -4/3, -5/3]",
      "relation": "implies",
      "rule_id": "System_CRAMER_NONZERO_D_BOTH_fractions_and_uniqueness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "D非0才除，x=(ce−bf)/D、y=(af−cd)/D，正den約分且保/1，兩原row已回代。任兩解的差解齊次式，消去得DΔx=DΔy=0，所以差0而唯一；域是R，不是整數單位條件。",
        "en": "Divide ONLY for D≠0:x=(ce−bf)/D,y=(af−cd)/D.Reduce positive denominators,retain/1,and verify BOTH original rows.The difference of two solutions obeys the homogeneous equations;elimination gives DΔx=DΔy=0,hence uniqueness over R,not an integer-unit criterion."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-4/3",
      "-5/3"
    ]
  },
  "conclusion": {
    "canonical_value": "(-4/3, -5/3)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261003-advance-20261004-r06-system-linear-equations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-advance-20261004-r06-system-linear-equations-foundations/questions.jsonl",
      "batch_sha256": "1482e4d4f4926204693a79846ba7e7d847e4a3539335e7d8b21cb87ed8ec2ad8",
      "original_line": 85,
      "original_line_sha256": "e916c4cd86425105cc416cdc7dfb29655986e29f843ffe5907a83e74137a3bd8",
      "question_compact_sha256": "3404b18894eba0da8ac3789666cc260eacfd8e48b8c2aadcc9eef1bb02ba464b",
      "english_row_compact_sha256": "5c5984f32f96eaecd279ccc2423f8a263b3203ebf6453c186dc962d36d3bec37",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r06-system-linear-equations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-r06-system-linear-equations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-system-linear-equations": {
        "sha256": "62dc0d59fbb66d23a8ef47c7b2725a56713d925c7f2c8ef0a380c46552972403",
        "english_sha256": "7fc6d87498d9c2a7943b1715c26c7d3e2baf79d0f54d4beb9019b628da0010c4",
        "original_path": "public/data/mko/system-linear-equations.json",
        "english_path": "public/data/i18n/editions/mko-system-linear-equations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470",
    "original_candidate_sha256": "65a6c0d717d9bd302cdd9ee2e7d07d1458c757457a4b4f206e86ec6c377f75d8",
    "original_candidate_line_sha256": "65a6c0d717d9bd302cdd9ee2e7d07d1458c757457a4b4f206e86ec6c377f75d8"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-011-p65",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b",
      "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 25,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "06344adf346719112bca18196b1fd6dcc34aa747be7f2c47dcb9b5a5bd9f48cf",
      "arithmetic_steps": 19
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}